Rca1948FrequencyModu.. - The New Jersey Antique Radio Club

302 FREQUENCY MODULATION, Volume Icos (a; cos 8) — J (x)00+2 2 (— l) n ^2«(z) cos2«0n=looand sin (x cos 0) = 2 2 (— l) B/ 2B+1 (a;) cos (2re + 1) 0.n=0Multiply the second equation by i and add the first equation. Changethe cosine terms to the complex form.This givescos (x cos 0) + i sin (x cos $)= J (*) + 2 (-l)»/ 2n (a;)e !!i »»+ 2 (- 1) "/»(*)«-""'+ » 2 (-lW2„+i(»0 e.< 2»+«« + i 2 (-l) B / 2 n+i(a!)e-«» +1)W .Since (—l)» = i 2 » = r- a » and /„(a:) = (— 1)" •/_„(«), this proves thelemma.Lemma j3sin cos (a cos 6 + b)= 2cos& 2 (-l) m Jzm+iia) sin (2m + 1) 8m=0In complex form, using lemma a2m+ 2sin&ec2 (— !)"• ^(a) sin2mfl.m=lasin cos (a cos 8 + b)1= (q fc=— oo_ e -tW-»>2 VJ k(a)eW — e -U8+v 2 (— * ^* (°) «** \k=— ook=—ooJl r «.= J 2 WkW [e» —4tL fc =--Me«'«+